| L(s) = 1 | − 3·2-s + 3-s + 7·4-s − 3·6-s + 2·7-s − 15·8-s + 3·9-s + 8·11-s + 7·12-s + 6·13-s − 6·14-s + 30·16-s + 2·17-s − 9·18-s + 10·19-s + 2·21-s − 24·22-s + 23-s − 15·24-s − 18·26-s + 14·28-s + 3·31-s − 57·32-s + 8·33-s − 6·34-s + 21·36-s − 3·37-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.577·3-s + 7/2·4-s − 1.22·6-s + 0.755·7-s − 5.30·8-s + 9-s + 2.41·11-s + 2.02·12-s + 1.66·13-s − 1.60·14-s + 15/2·16-s + 0.485·17-s − 2.12·18-s + 2.29·19-s + 0.436·21-s − 5.11·22-s + 0.208·23-s − 3.06·24-s − 3.53·26-s + 2.64·28-s + 0.538·31-s − 10.0·32-s + 1.39·33-s − 1.02·34-s + 7/2·36-s − 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.992288518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.992288518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| good | 2 | $C_2^2:C_4$ | \( 1 + 3 T + p T^{2} + T^{4} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 5 p T^{5} - 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4\times C_2$ | \( 1 - 8 T + 13 T^{2} + 74 T^{3} - 435 T^{4} + 74 p T^{5} + 13 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 6 T + 23 T^{2} - 120 T^{3} + 601 T^{4} - 120 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 2 T + 7 T^{2} - 70 T^{3} + 441 T^{4} - 70 p T^{5} + 7 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 10 T + 21 T^{2} + 70 T^{3} - 469 T^{4} + 70 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 - T + 28 T^{2} - 65 T^{3} + 501 T^{4} - 65 p T^{5} + 28 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 19 T^{2} + 120 T^{3} + 721 T^{4} + 120 p T^{5} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 3 T - 22 T^{2} + 159 T^{3} + 205 T^{4} + 159 p T^{5} - 22 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 155 p T^{5} - 18 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 2 T - 17 T^{2} + 214 T^{3} + 2025 T^{4} + 214 p T^{5} - 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 3 T - 43 T^{2} - 45 T^{3} + 2116 T^{4} - 45 p T^{5} - 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 11 T + 8 T^{2} - 65 T^{3} + 3011 T^{4} - 65 p T^{5} + 8 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 15 T + 31 T^{2} + 15 p T^{3} - 10424 T^{4} + 15 p^{2} T^{5} + 31 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 17 T + 78 T^{2} + 289 T^{3} + 3755 T^{4} + 289 p T^{5} + 78 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 12 T - 3 T^{2} + 850 T^{3} - 7779 T^{4} + 850 p T^{5} - 3 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 3 T - 37 T^{2} + 549 T^{3} + 1480 T^{4} + 549 p T^{5} - 37 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 9 T + 8 T^{2} - 585 T^{3} - 5849 T^{4} - 585 p T^{5} + 8 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 10 T + 21 T^{2} + 770 T^{3} + 12791 T^{4} + 770 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - T - 52 T^{2} - 565 T^{3} + 6761 T^{4} - 565 p T^{5} - 52 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 - 20 T + 151 T^{2} - 1600 T^{3} + 21441 T^{4} - 1600 p T^{5} + 151 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 7 T - 63 T^{2} - 185 T^{3} + 11276 T^{4} - 185 p T^{5} - 63 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.67802960919646389333371473405, −7.41707998820222148555908196622, −7.30673839466579759375087409742, −6.97129116628735182148506910590, −6.88400677695964226748609894547, −6.52783574932759479109328076942, −6.52243909605835426566798231633, −5.97543184889933885361169243350, −5.84421863164464804575837053788, −5.76476939896673793063474053548, −5.44284442383642862502591924466, −4.96554661938319594675914135405, −4.69566187999585586739173617414, −4.21063844838221873955999251520, −4.10787357752464197156482770503, −3.50148338704231002024084849901, −3.31129163823455586109862962696, −3.28452001713052722585969051794, −3.16952633195153945371273272404, −2.39848056517299257492800654530, −1.85452047759304657508849830616, −1.85002432249993949617277672156, −1.27961956511686875622075168495, −1.16163044505514384670249480912, −0.844458708252961436927600153839,
0.844458708252961436927600153839, 1.16163044505514384670249480912, 1.27961956511686875622075168495, 1.85002432249993949617277672156, 1.85452047759304657508849830616, 2.39848056517299257492800654530, 3.16952633195153945371273272404, 3.28452001713052722585969051794, 3.31129163823455586109862962696, 3.50148338704231002024084849901, 4.10787357752464197156482770503, 4.21063844838221873955999251520, 4.69566187999585586739173617414, 4.96554661938319594675914135405, 5.44284442383642862502591924466, 5.76476939896673793063474053548, 5.84421863164464804575837053788, 5.97543184889933885361169243350, 6.52243909605835426566798231633, 6.52783574932759479109328076942, 6.88400677695964226748609894547, 6.97129116628735182148506910590, 7.30673839466579759375087409742, 7.41707998820222148555908196622, 7.67802960919646389333371473405
